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In this letter, we demonstrate a new five-element true-time-delay (TTD) system that uses only a single tunable linear chirped fiber grating (TLCFG) delay line for continuous TTD beamforming. The system avoids the complicated wavelength tuning and synchronal control of the tunable lasers as well as the tunable bandpass filters. Different time delays are achieved by tuning the chirp rate of the TLCFG, to which a multiwavelength laser source that carries the RF signal is applied. The time delay responses of the TTD system are measured experimentally, which agree well with the theoretical analysis. The beampointing angle can be controlled continuously by changing the chirp rate of the TLCFG. The minimum time delay step can be much smaller than 1 ps thanks to the high position resolution of the actuator.